Unitary Symmetries

A unitary symmetry is a symmetry represented on Hilbert space by a unitary operator $U$. When all elements of a symmetry group are assigned compatible unitary operators, this is a unitary representation:

$$U^\dagger U=UU^\dagger=I.$$

Unitary symmetries include spatial translations, rotations, parity, internal phase symmetries, and many finite symmetry operations. They are the default symmetry operations in ordinary quantum mechanics, with antiunitary symmetries treated separately.

Preservation of Inner Products

A unitary operator preserves inner products:

$$\langle U\phi|U\psi\rangle = \langle\phi|\psi\rangle.$$

It therefore preserves norms and transition probabilities:

$$\lvert\langle U\phi|U\psi\rangle\rvert^2 = \lvert\langle\phi|\psi\rangle\rvert^2.$$

This is why unitary operators are natural representatives of quantum symmetries on state vectors.

State and Operator Transformations

Under an active unitary transformation,

$$\lvert\psi\rangle\longmapsto U\lvert\psi\rangle.$$

Operators transform as

$$A\longmapsto UAU^\dagger.$$

If both states and operators are transformed consistently, expectation values are unchanged:

$$\langle U\psi|UAU^\dagger|U\psi\rangle = \langle\psi|A|\psi\rangle.$$

This invariance is a statement about representing the same physical relation after the transformation.

Symmetry of a Hamiltonian

A unitary transformation is a symmetry of a Hamiltonian when

$$UHU^\dagger=H.$$

Equivalently,

$$[U,H]=0.$$

When this holds, $U$ maps solutions of the Schrödinger equation to solutions with the same energy structure. If $\lvert E\rangle$ is an energy eigenstate, then $U\lvert E\rangle$ is also an energy eigenstate with the same energy.

Continuous Unitary Symmetries

A one-parameter unitary group is commonly written

$$U(\alpha)=e^{-i\alpha G/\hbar},$$

where $G$ is a Hermitian generator. If $U(\alpha)$ is a symmetry for all $\alpha$, then

$$[G,H]=0.$$

This is the quantum symmetry-conservation link. Examples include:

• translations generated by momentum,
• rotations generated by angular momentum,
• time translations generated by the Hamiltonian.

Examples

Translations in one dimension:

$$U(a)=e^{-iaP/\hbar}.$$

A free-particle Hamiltonian

$$H=\frac{P^2}{2m}$$

commutes with $U(a)$ for all $a$.

Rotations, with their spatial group described by $SO(3)$, are represented by

$$U(\hat{\mathbf n},\theta) = \exp\left( -\frac{i}{\hbar}\theta\,\hat{\mathbf n}\cdot\mathbf J \right).$$

A rotationally invariant Hamiltonian commutes with all components of $\mathbf J$.

Parity in one dimension is unitary and satisfies

$$P^2=I.$$

It is a symmetry when the Hamiltonian is invariant under $x\mapsto -x$.

Common Mistakes

• Treating every unitary operator as a symmetry of a specific Hamiltonian.
• Forgetting that unitary transformations preserve all inner products, not only norms.
• Confusing active transformations with changes of basis.
• Writing a continuous symmetry without identifying its generator.
• Assuming a symmetry generator is conserved when the Hamiltonian has explicit symmetry-breaking terms.

Cross-Links

• Quantum Symmetries
• Active and Passive Transformations
• States, Observables, and Hamiltonians
• Symmetry Groups and Representations
• One-Parameter Unitary Groups
• Symmetries and Dynamical Automorphisms
• Wigner’s Theorem Preview
• Antiunitary Symmetries
• Antiunitary Symmetries, First Look
• Symmetry Constraints on Hamiltonians
• Exact Symmetry
• Groups
• Group Actions
• Unitary Representations
• SO(3)
• Lie Groups
• Unitary Time Evolution
• Hamiltonians as Generators
• Commutator Table

References

• J. J. Sakurai and J. Napolitano, Modern Quantum Mechanics, 3rd ed., Cambridge University Press, 2020.
• R. Shankar, Principles of Quantum Mechanics, 2nd ed., Springer, 1994.
• B. C. Hall, Quantum Theory for Mathematicians, Springer, 2013.

Exercises

1. Show that a unitary operator preserves transition probabilities.

Solution

Using $U^\dagger U=I$,

$$\langle U\phi|U\psi\rangle = \langle\phi|U^\dagger U|\psi\rangle = \langle\phi|\psi\rangle.$$

Taking the absolute square gives preservation of transition probabilities.

2. If $[U,H]=0$ and $H\lvert E\rangle=E\lvert E\rangle$, show that $U\lvert E\rangle$ is also an energy eigenstate with energy $E$.

Solution

Since $HU=UH$,

$$H(U\lvert E\rangle) = UH\lvert E\rangle = E(U\lvert E\rangle).$$

Thus $U\lvert E\rangle$ is either zero or an eigenstate with the same energy. A unitary operator cannot map a nonzero vector to zero.